The area of a triangle is $706$ . Two of the side lengths are $45$ and $33$ and the included angle is obtuse. Find the measure of the included angle, to the nearest tenth of $a$ degree. $\newline$ Answer:

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Q. The area of a triangle is

706

. Two of the side lengths are

45

and

33

and the included angle is obtuse. Find the measure of the included angle, to the nearest tenth of

a

degree.

\newline

Answer:

Calculate Area Formula: To find the measure of the included angle, we can use the formula for the area of a triangle when two sides and the included angle are known: $\text{Area} = \frac{1}{2} \cdot a \cdot b \cdot \sin(C)$ , where $a$ and $b$ are the side lengths, and $C$ is the included angle. $\newline$ Given that the area is $706$ square units, and the side lengths are $45$ and $33$ units, we can set up the equation: $\newline$ $706 = \frac{1}{2} \cdot 45 \cdot 33 \cdot \sin(C)$
Set Up Equation: Now, we need to solve for $\sin(C)$ . First, we multiply $45$ and $33$ , and then divide $706$ by the result, and finally multiply by $2$ to isolate $\sin(C)$ on one side of the equation: $\newline$ $\sin(C) = \frac{2 \times 706}{45 \times 33}$
Solve for sin(C): Perform the calculations to find the value of sin(C): $\newline$ $\sin(C) = \frac{2 \times 706}{45 \times 33} = \frac{1412}{1485} \approx 0.951$
Find Acute Angle: Since the angle is obtuse, it lies in the second quadrant where sine is positive. We need to find the angle whose sine is approximately $0.951$ . We use the inverse sine function to find the angle, but since the range of $\arcsin$ is from $-\pi/2$ to $\pi/2$ , we need to find the supplement of the angle that $\arcsin$ would give us because the angle is obtuse. $\newline$ Let's first find the acute angle whose sine is $0.951$ : $\newline$ $C_{\text{acute}} = \arcsin(0.951)$
Calculate Acute Angle: Calculate the acute angle using a calculator: $C_{\text{acute}} \approx \arcsin(0.951) \approx 71.8$ degrees
Find Obtuse Angle: To find the obtuse angle, we take the supplement of the acute angle: $C_{\text{obtuse}} = 180^\circ - C_{\text{acute}}$
Calculate Obtuse Angle: Perform the calculation to find the measure of the obtuse angle: $C_{\text{obtuse}} = 180 \text{ degrees} - 71.8 \text{ degrees} \approx 108.2 \text{ degrees}$